I can do all of this question except the last part: Let $N$ be a normal subgroup of a finite group $G$ of prime index $p$. Show that if $H$ is a subgroup of $G$ then $H\cap N$ is a normal subgroup of $H$ of index $1$ or $p$. Could I have some help please?
I've shown that $H\cap N$ is a subgroup using the one-step test, and I've shown it's normal by conjugating it with an element of $H$.
I'm stuck on identifying the index of $H\cap N$. I'm guessing it involves Lagrange's Theorem, and I tried to come up with a homomorphism from $H$ to something with kernel $H\cap N$, but with no success.
I'd prefer hints to a full solution, if that's alright.
Thank you very much for all your help.